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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 29th 2011

    had need for a stub for local diffeomorphism

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 2nd 2015

    Suppose a smooth function p:X np \colon X \to \mathbb{R}^n from a diffeological space XX to Cartesian space induces at each point an isomorphism on tangent vectors as well as on all higher jets.

    Then what sensible extra conditions does it take to conclude that pp is in fact a local diffeomorphism, i.e. restricts to a diffeomorphism around an open neighbourhood of each point?

    Here I mean tangents and jets defined by equivalence classes of smooth maps into XX.

    • CommentRowNumber3.
    • CommentAuthorigor
    • CommentTimeMay 3rd 2015

    Unless I’m missing something, but your condition implies that the Jacobian map Tp:TXT nT p \colon TX \to T \mathbb{R}^n is full rank and even invertible. The inverse function theorem then guarantees that pp is a local diffeomorphism about any point xXx\in X that has an nn-manifold neighborhood. Is your question then about XX’s that at some points fail to be nn-manifold? But then, it seems to me, that essentially by definition there cannot be a local diffeomorphism from any neighborhood of such a point into n\mathbb{R}^n.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMay 5th 2015

    If you only know that XX is a diffeological space and that it has a map X nX \to \mathbb{R}^n which is an iso on all tangents, what else does it need (if anything) to conclude that XX is a manifold sitting by a local diffeomorphism over n\mathbb{R}^n?

    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeMay 5th 2015
    • (edited May 5th 2015)

    What happened with moneomorphism and epiomorphism? I know that the terms are used rarely, especially outside of Eastern Europe, and especially the second term, but I remember writing about that terminology in nlab and this seemingly completely vanished. nLab search and google show no hits at nLab about those. Did I dream about writing it ?

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